Chapter 3, Problem 42RE

Solution

To graph: The function and analyse it.

Exact coordinates of absolute minimum point $\left(e,\frac{1}{e}\right)$

Given information: The function is $f(x)=\frac{\ln x}{x}$ .

Calculation:

The graph $f(x)$ is indicated below:

  

Now we should analyse function based on our observation of graph:

Domain: all real $x<0$ .

Range: all real $y\le -0.368$ .

Continuity: continuous on its domain.

Increasing or decreasing behaviour: decreasing on $\left(0,-2.718\right)$ increasing on $\left(2.718,\infty \right)$ .

Symmetry: none

Boundedness: bounded below $y=0.368$ .

Extreme: absolute maximum $\left(\mathrm{2.718.}-0.368\right)$

Asymptotes: none

End behaviour:

  $\begin{array}{l}\underset{x\to \infty }{\lim }f(x)=0\\ \underset{x\to 0^{+}}{\lim }f(x)=-\infty \end{array}$

Exact coordinates of absolute minimum point found are $\left(e,\frac{1}{e}\right)$ .

But this cannot be proven without calculus we use the decimal approximation as found in graphing utility.